Method to determine a representative parameter of a porous sample and related system

ABSTRACT

The method comprises feeding a second fluid in a porous sample; measuring a resistivity or/and conductivity in a plurality of regions having different second fluid contents in the porous sample; and repeating the following steps. Determining an estimated local volume of first fluid contained in each region from the resistivity or/and conductivity measured in the region and from an estimated value of the representative parameter; calculating an estimated total volume of first fluid in the porous sample from each estimated local volume of first fluid contained in each region; and modifying the value of the estimated representative parameter to minimize the difference between the estimated total volume and a measured total volume of fluid produced from the porous sample, the representative parameter of the porous sample being the estimated representative parameter minimizing said difference.

CROSS REFERENCE TO RELATED APPLICATION

The present application is a U.S. National Phase Application under 35U.S.C. § 371 of International Patent Application No. PCT/IB2020/000079filed Jan. 23, 2020. The entire contents of which are herebyincorporated by reference.

FIELD

The present invention concern a method for determining a representativeparameter of a porous sample in an equation relating the resistivityor/and conductivity of the porous sample, with a saturation of theporous sample in a first fluid, the method providing a porous samplecontaining a first fluid; feeding a second fluid in the porous sampleand establishing at least a first profile of second fluid content in theporous sample by applying a first mechanical load; measuring aresistivity or/and conductivity in a plurality of regions havingdifferent second fluid contents in the porous sample.

BACKGROUND

Such a method is used for example to determine the exponent coefficientn of the brine saturation S_(w) in Archie's law. The porous sample isfor example a rock sample recovered from a sub-soil formation.

When drilling a well, it is known to recover solid samples from theformations through which the well is drilled, in particular rocksamples.

In the same well, measurement tools can be conveyed in order to performvarious measurements at various depths. This process is called logging.The result of this process is a continuous measurement of properties(porosity, resistivity . . . ) as a function of depth.

Rock samples are used to measure properties in laboratory and tocalibrate the logs.

The log is generally obtained by visual inspections of the samplesrecovered at the surface, and/or by physical measurements carried outalong the well.

In a logging operation, electrical conductivity is often measured.Electrical conductivity can be related to significant parameters of theformations, including in particular porosity and saturation.

For example, an empirical law such as Archie's law relates theelectrical conductivity of a porous sample of formation to its porosityand to its brine saturation. In a fluid saturated rock, the brinesaturation is then related to hydrocarbon saturation, providingextremely relevant information about the location and potential ofhydrocarbon reservoirs after the well is drilled.

Archie's law, reformulated for electrical resistivity reads as follows:

R _(t) =a×Φ ^(−m) ×S _(w) ^(−n) ×R _(w)

-   -   in which R_(t) is the sample resistivity, R_(w) is the        resistivity of the brine (which depends on salinity and        temperature), ϕ is the sample porosity, and a is a constant.

The formation factor a×ϕ^(−m) is related to the resistivity of theporous sample saturated only with brine by the equationR₀=a×ϕ^(−m)×R_(w). Consequently, a resistivity index RI can bedetermined following the following equation:

${\log RI} = {{\log\left( \frac{R_{t}}{R_{0}} \right)} = {{- n} \times \log S_{w}}}$

In order to use Archie's law, the exponent coefficient n associated withthe brine saturation S_(w) has to be experimentally determined for aparticular porous sample.

Experimental determination of Archie's law exponent coefficient n isgenerally a long and tedious process.

A porous sample containing water is inserted in a cell. Oil underpressure is injected in the porous sample, at one end of the poroussample, and another end of the porous sample is equipped with a porousplate from which only water is able to be extracted.

After a long time, generally in the order of a month, a steady state isreached in the porous sample. An average saturation S_(w) in water ofthe porous sample is measured.

In parallel, the resistivity R_(t) of the porous sample is measured byplacing electrodes at the ends of the porous sample when the steadystate is reached.

A first point of the curve connecting the logarithm of the saturationwith the resistivity index is thus obtained. The capillary pressure ishere equal to the pressure at which the oil is injected and a firstpoint of the curve of the capillary pressure versus saturation isobtained.

Then, the oil pressure is increased at the porous sample inlet. Theexperiment is stayed until a steady state is reached. When the steadystate is reached, a second measurement of the resistivity and of thecapillary pressure is carried out to obtain a second point of the abovementioned curve.

The previously described operations must then be repeated several timesuntil an adequate number of points is determined.

Consequently, the measurement of the determination of the exponentcoefficient n of Archie's law and of the pressure of the capillarypressure Pc versus saturation takes several months. This significantlydelays the log interpretation and the resultant business decisions forthe operations.

In order to speed up the experimental determination of the Archie's lawcoefficient, WO 2018/193282 discloses a method of the above-mentionedtype, in which a steady state profile of a second fluid is establishedin the porous sample by centrifugation.

After the steady state is obtained, the porous sample is extracted fromthe centrifuge and a measurement of resistivity and of water saturationis carried out in a plurality of regions along the porous sample. Basedon the corresponding values of local resistivity and water saturationobtained in each region, a correlation is made to determine Archie's lawexponent.

Such a method is much faster than the traditional measurement process.However, it still requires a lot of sample handling, first to load thesample in the centrifuge, then to unload the sample from the centrifuge,and thereafter, to transfer the sample to a resistivity measurementapparatus and to a saturation measurement apparatus such as a nuclearmagnetic resonance system.

The need for measuring the sample in at least two very different devicesfor obtaining resistivity and saturation values also delay the provisionof the results.

SUMMARY

One aim of the invention is to obtain a robust method for determining arepresentative parameter of a porous sample and associated relationshipscorrelating physical quantities of the porous sample, the method beingvery fast to operate, with minimal sample handling.

To this aim, the subject matter of the invention is a method of theabove-mentioned type, characterized by:

-   -   measuring a volume of first fluid produced from the porous        sample when establishing the first profile;    -   calculating a measured total volume of first fluid remaining in        the porous sample after establishing the first profile; and    -   repeating the following steps:        -   determining an estimated local volume of first fluid            contained in each region from the resistivity or/and            conductivity measured in the region and from an estimated            value of the representative parameter;        -   calculating a estimated total volume of first fluid in the            porous sample from each estimated local volume of first            fluid contained in each region;        -   modifying the value of the estimated representative            parameter to minimize the difference between the estimated            total volume and the measured total volume,        -   the representative parameter of the porous sample being the            estimated representative parameter minimizing said            difference.

The method according to the invention may comprise one or more of thefollowing feature(s), taken solely, or according to any technicalfeasible combination:

-   -   the first profile is a first steady state profile;    -   the local estimated volume of first fluid contained in each        region is calculated from a saturation in first fluid obtained        from an inversion of the equation relating the resistivity        or/and conductivity and the saturation in a first fluid in the        porous sample, using the resistivity or/and conductivity        measured in the region and using the estimated value of the        representative parameter;    -   determining the local estimated volume of first fluid contained        in each region comprises calculating a local volume of pores in        each region, advantageously from a global volume of pores        determined in the porous sample;    -   it comprises, after measuring a resistivity or/and conductivity        in a plurality of regions having different second fluid contents        in the porous sample:        -   establishing at least one subsequent profile of second fluid            content in the porous sample by applying a subsequent            mechanical load different from the first mechanical load;        -   measuring in the plurality of regions a subsequent            resistivity and/or conductivity and a subsequent volume of            first fluid collected in the establishment of the subsequent            profile,        -   calculating a remaining volume of first fluid in the porous            sample,    -   the repeating step comprising, after each profile has been        established:        -   determining an estimated volume of first fluid contained in            each region from the resistivity or/and conductivity            measured in the region and from the estimated value of the            representative parameter;        -   calculating an estimated total volume of first fluid in the            porous sample from each determined estimated local volume of            first fluid volume contained in each region after each            profile has been established;        -   modifying the value of the estimated representative            parameter to minimize the difference between each estimated            total volume and the corresponding measured total volume            after each profile has been established, the representative            parameter of the porous sample being the estimated            representative parameter minimizing said difference;    -   the or each subsequent profile is a subsequent steady state        profile;    -   the first profile and/or the or each subsequent profile are        established in a centrifuge.    -   establishing the first profile and the or each subsequent        profile is carried out without removing the porous sample from        the centrifuge;    -   minimizing the difference between the estimated cumulated volume        and the measured volume comprises calculating a cost function        according to the following equation:

$\sum\limits_{j = 1}^{Nj}\left\lbrack {{\sum\limits_{i = 1}^{Ne}\left\{ {\left( 10^{\frac{\log(\frac{{Rt}({i,j})}{R0})}{- n}} \right) \times {V_{{po},{est}}(i)}} \right\}} - {{VT}_{1,{mes}}(j)}} \right\rbrack^{2}$

in which Nj is the total number of steps of establishing a profile ofsecond fluid in the porous sample, Ne is the total number of regions inthe porous sample, R_(t)(i,j) is a resistivity in a i^(th) region aftera j^(th) step of establishing a profile of second fluid in the poroussample, R0 is an initial resistivity in a i^(th) region before feedingthe second fluid in the porous sample, n is an estimated representativeparameter, V_(po,est) is an estimated volume of pores in the i^(th)region, VT_(1,mes)(j) is a measured volume of first fluid inside theporous sample 10 after a j^(th) step of establishing a profile of secondfluid in the porous sample.

-   -   measuring a resistivity or/and conductivity in a plurality of        regions having different second fluid contents in the porous        sample is carried out while the first mechanical load is        applied;    -   the equation relating the resistivity or/and conductivity and        the saturation in a first fluid is chosen among Archie's law,        Waxman-Smits's law, Poupon-Leveaux's law, Simandoux's law,        Clavier-Coates Dumanoir's Dual-Water law and/or the effective        Spalburg's medium model law;    -   the first fluid is a water based fluid in particular brine, the        second fluid being an oil based fluid;    -   the porous sample is a formation sample, in particular a rock        sample;    -   it comprises, before feeding a second fluid in the porous sample        and establishing a first profile of the second fluid, measuring        a resistivity or/and conductivity in the plurality of regions        when the porous sample only contains the first fluid.

The invention further a system for determining a representativeparameter of a porous sample in an equation relating the resistivityor/and conductivity of the porous sample, with a saturation of theporous sample in a first fluid, the system comprising:

-   -   a cell for receiving a porous sample containing a first fluid;    -   an apparatus for feeding a second fluid in the porous sample and        for establishing at least a first profile of second fluid        content in the porous sample by applying a first mechanical        load;    -   resistivity or/and conductivity measuring sensors for measuring        a resistivity or/and conductivity in a plurality of regions        having different second fluid contents in the porous sample;

characterized by:

-   -   a fluid production sensor for measuring a volume of first fluid        produced from the porous sample when establishing the first        profile;    -   a calculator configured to calculate a measured total volume of        first fluid remaining in the porous sample after establishing        the first profile; and

the calculator being configured to repeat the following steps:

-   -   determining an estimated volume of first fluid contained in each        region from the resistivity or/and conductivity measured in the        region and from an estimated value of the representative        parameter;    -   calculating a estimated total volume of first fluid in the        porous sample from each determined estimated local volume of        first fluid volume contained in each region;    -   modifying the value of the estimated representative parameter to        minimize the difference between the estimated total volume and        the measured total volume, the representative parameter of the        porous sample being the estimated representative parameter        minimizing said difference.

The system according to the invention may comprise the followingfeature:

-   -   the apparatus for feeding a second fluid in the porous sample        and for establishing a first profile of second fluid content in        the porous sample by applying a first mechanical load comprises        a centrifuge having a static part and a rotor rotatable with        regards to the static part, the resistivity or/and conductivity        measuring sensors being electrically powered sensors carried by        the rotor, the centrifuge comprising a contactless power and        signal transmission system between the rotor and the static part        of the centrifuge.

Advantageously, the first profile is a first steady state profile.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be better understood, upon reading of the followingdescription, given only as an example, and made in reference to thefollowing figures, in which:

FIG. 1 is a schematic view of a system for carrying out a methodaccording to the invention;

FIG. 2 is a detail of FIG. 1 showing a contactless power and signaltransmission system of the centrifuge in the system of FIG. 1 ;

FIG. 3 is a schematic view of a cell containing a porous sample;

FIG. 4 is a schematic view of a sensor system comprising a plurality ofresistivity probes in a preliminary step of the method according to theinvention;

FIG. 5 is a view similar to FIG. 4 , in a further step of the methodaccording to the invention.

DETAILED DESCRIPTION

A method for determining a representative parameter of a porous sample10 shown in FIG. 3 is carried out in the measuring system 12schematically illustrated on FIGS. 1 and 2 .

The representative parameter is a parameter in an equation relating atleast two physical quantities associated with the porous sample 10, suchas resistivity and/or conductivity on the one hand, and saturation in afirst fluid of the porous sample 10, on the other hand.

Preferably, the representative parameter is the exponent saturationcoefficient n in an empirical equation relating conductivity and/orresistivity to saturation in a first fluid of the porous sample 10.

The equation is for example Archie's law as defined above. In a variant,the equation is chosen among Waxman-Smits's law, Poupon-Leveaux's law,Simandoux's law, Clavier-Coates Dumanoir's Dual-Water law and/or theeffective Spalburg's medium model law.

Simultaneously to determining the representative parameter, the methodaccording to the invention advantageously allows a determination of therelationship relating capillary pressure to saturation in a first fluidfor the porous sample 10.

The porous sample 10 is for example a formation sample extracted from asub-soil. The formation sample is in particular a rock sample having aninternal porosity.

Typically, the porous sample 10 has for example a volume comprisedbetween 8 cm³ and 60 cm³. It is advantageously cylindrical, with acircular cross-section.

The diameter of the porous sample 10 is generally comprised between 23mm and 40 mm. Its length is for example comprised between 20 mm and 50mm.

In a variant, the porous sample 10 is a parallelepiped.

The measuring system 12 comprises a cell 14 receiving the porous sample10 filled with a first fluid (see FIG. 3 ), and an apparatus 16 forestablishing at least a profile, in particular a steady state profile,of a second fluid content in the porous sample 10 by applying a firstmechanical load and for measuring a volume Vp of first fluid producedfrom the porous sample 10, when establishing the profile.

As shown in FIG. 5 , the apparatus 16 is thus able to create, in theporous sample 10, a plurality of regions 17 having different secondfluid contents in the porous sample 10 and to measure, in each of theplurality of regions 17, a corresponding local electrical resistivityR_(t) (i) and/or conductivity C_(t)(i).

The measuring system 12 further comprises a calculator 22 for estimatinga value of the total volume VT_(1,est) of first fluid in the poroussample 10 from the local resistivities R_(t) (i) and/or conductivitiesC_(t)(i) measured in each region 17 and for determining therepresentative parameter n, based on minimizing the difference betweenthe estimated volume VT_(1,est) and a measured volume VT_(1,mes) offirst fluid in the porous sample 10 obtained from the measured volume Vpof first fluid produced from the porous sample 10 when establishing thesteady state profile.

An example of cell 14 is shown schematically in FIG. 3 . It comprises aclosed enclosure 30 defining a volume 32 for receiving the porous sample10, an upstream chamber 34, for injection of the second fluid in theporous sample 10, and a downstream chamber 36 for receiving fluidscollected when a mechanical load is applied to the porous sample 10.

The cell 14 delimits at least an inlet 38 for feeding the second fluidinto the upstream chamber 34. It extends along a longitudinal axis X-X′which is coaxial with the longitudinal axis of the porous sample 10.

The inlet 38 is able to be closed to seal the enclosure 30. Chambers 34and 36 are able to fluidly communicate to equilibrate pressures whenfluid is produced from the porous sample 10 in either of the chambers34, 36 as will be described below.

The cell 14 defines at least a transparent window in the downstreamchamber 36 and/or in the upstream chamber 34.

Advantageously, the enclosure 30 of the cell 14 comprises an assembly ofa centrifuge cup containing the porous sample 10 and of a transparenttest tube delimiting the downstream chamber 36.

As shown in FIGS. 1 and 2 , the apparatus 16 comprises a centrifuge 40,a sensing control system 42 and a control unit 44.

The centrifuge 40 comprises an outer enclosure 46 defining an innervolume 48, and a centrifuge rotor 50 equipped with electrically poweredsensors 52 to measure at least a property of the porous sample inregions 17 of the porous sample.

The centrifuge 40 further comprises a motor 54 able to drive the rotor50 in rotation around a rotation axis A-A′, a hood 56 able to close theinner volume 48, and an electrical power source 57 able to power themotor 54.

The centrifuge 40 also comprises a contactless power and signaltransmission system 58 able to transmit power from the source 57 to theelectrically powered sensors 52 and to transmit information from theelectrically powered sensors 52 to the control unit 44 during rotationof the rotor 50 around the rotation axis A-A′. It further comprises anelectrical connection 59 connecting the electrically powered sensors 52to the contactless power and signal transmission system 58.

The enclosure 46 and the hood 56 remain static in rotation around therotation axis A-A′ when the rotor 50 rotates around the rotation axisA-A′. They will be referred to as the “static” part of the centrifuge 40in the following description. More generally, the term “static” shouldbe understood as static in rotation around the axis A-A′ when the rotor50 rotates. It does not prevent a movement to occur, for example of thehood 56 with regards to the enclosure 46 to allow access to the innervolume 48.

The centrifuge rotor 50 comprises a support 60 defining at least ahousing 62 for receiving a porous sample 10 housed in a cell 14.

The support 60 also holds the electrically powered sensors 52 and theelectrical connection 59, at least when the cell 14 is received in thehousing 62.

In the example shown in FIG. 1 , the support 60 comprises a bowl 64,mounted rotatable in the inner volume 48 around the axis A-A′, a centralhub 66 connecting the motor 54 to the bowl 64 and a rotatable holder 68for holding a first part of the contactless power and signaltransmission system 58.

The bowl 64 has a bottom wall 70 and a lateral wall 72 defining acentral cavity 74 around axis A-A′. The bowl 64 is driven by the motor54 to rotate around axis A-A′ at a speed comprised between 0 rpm and15000 rpm.

The central cavity 74 receives at least a frame 76 defining each housing62. The frame 76 preferentially comprise arms with protrude radiallyfrom the central hub 66.

Each housing 62 extends radially with regards to the rotation axis A-A′.When received in the housing 62, in particular during rotation of therotor 50, the axis X-X′ of each porous sample 10 extends radially withregards to the rotation axis A-A′.

The electrically powered sensors 52 are mounted in the cell 14 aroundthe sample 10. In the example of FIG. 4 , the sensors 52 comprise atleast two plate electrodes 80, mounted at the ends of the porous sample10 and intermediate electrodes 82 each formed of at least a coil ofwire, which are distributed along the length of the porous sample 10.

Each plate electrode 80 and an adjacent electrode 82, and each electrode82 and another adjacent electrode 82 define between them a successiveregion 17 of the porous sample 10 in which a local resistivity R_(t)(i)and/or conductivity C_(t)(i) is measured by determining the tensionarising between two successive adjacent electrodes 80, 82, or 82 when acurrent circulates in the sample 10. It may comprise a supplementaryradial electrode.

In the example of FIGS. 4 and 5 , the successive regions 17 are slicesof the porous sample 10 taken in succession longitudinally along thelength of the porous sample 10. Each slice is delimited by two paralleltransverse planes which are perpendicular to the longitudinal axis X-X′of the porous sample 10. The number of regions 17 is for examplecomprised between 4 and 15 preferably between 7 and 11. The length ofeach region 17, taken along the axis is preferably smaller than 20% ofthe total length of the porous sample 10.

Advantageously, the electrically powered sensors 52 comprise a furthersensor 84 measuring the sealing of the cell 14, for example by aresistivity measurement at the interface of the cell 14 to check that noleak occurs during centrifugation.

The electrical connection 59 comprise wires or leads connecting theelectrically powered sensors 52 to the contactless power and signaltransmission system 58 along the frame 76.

For example, the electrical connection 59 has at least a conductor witha first section running radially along the arms and a second sectionwhich runs axially to the rotatable holder 68.

The rotatable holder 68 preferentially has a surface which extendsperpendicular to the rotation axis A-A′.

The motor 54 of the centrifuge 40 is able to be actuated by the controlunit 44 to rotate the rotor 50 and jointly the cell 14 containing theporous sample 10 and the electrically powered sensors 52 at a speed ofrotation ranging from 0 rpm to 15000 rpm.

A mechanical load in the form of a centrifugal force applies on theporous sample 10 and on the fluid contained in the porous sample 10.This leads to impregnation of the porous sample 10 with the second fluidcontained in the upstream chamber 34 and to first fluid production inthe downstream chamber 36.

The power source 57 is for example an electrical connection to anelectrical network or to a generator.

The sensing unit 42 comprises a rotation speed sensor 90 able to detectthe speed of rotation of the rotor 50 and a fluid production sensor 92.

The fluid production sensor 92 is able to monitor the rate of fluidproduction of the fluid sample 10 during rotation of the cell 14 aroundthe rotation axis A-A′. In the example shown in FIGS. 1 and 2 , thefluid production sensor 92 comprises at least a stroboscope 92B and acamera 92A able to take images of the content of the downstream chamber36 and/or of the upstream chamber 34 along time.

The control unit 44 is able to analyze the fluid production from theimages taken in the camera 92A and to relate it to a rate of productionof fluid and to a volume of produced fluid V_(p) in the downstreamchamber 36 and/or in the upstream chamber 34 by image analysis.

The hood 56 comprises a door 86 able to close the inner volume 48 of theenclosure 46, and a stand 87 for receiving a second part of thecontactless power and signal transmission system 58.

In this example, the hood 56 has a trough opening 86A provided throughthe door 86 to let the camera 92A take images of the content of thedownstream chamber 36 and/or of the upstream chamber 34 along time.

In the example shown in FIG. 2 , the stand 87 comprises a lower staticplate 88 which protrudes transversally and perpendicularly to axis A-A′,when the door 86 closes the inner volume 48.

In that position, the lower plate 88 faces the rotatable holder 68 ofthe centrifuge rotor 50, parallel to the upper surface of the holder 68.

An air gap 89 is defined between the lower plate 88 and the rotatableholder 68. The gap 89 for example has a height of at least 0.1 mm, inparticular comprised between 1 mm and 20 mm.

Thus, the rotatable holder 68 is able to rotate coaxially to the lowerplate 88 around the rotation axis A-A′, facing the lower plate 88,without contact with the lower plate 88.

As shown in FIG. 2 , the contactless power and signal transmission 58comprises a static contactless power transmitter 100 held by the lowerplate 88 and a rotatable contactless power receiver 102 held by therotatable holder 68.

In addition, the contactless power and signal transmission system 58further comprises a rotatable contactless signal transmitter 104 held bythe holder 68 and a static contactless signal receiver 106 held by thelower plate 88.

The contactless power transmitter 100 comprises a static antenna made ofa coil of wires 108 and a first electronic card 110 able to injectelectrical power from the source 57 to the static coil of wires 108.

The contactless power receiver 102 comprises a rotatable antenna made ofa coil of wires 112, able to receive electrical power from the staticcoil of wires 108 by contactless power transmission and a secondelectronic card 114 able to receive electrical power from the rotatablecoil of wires 112 and to distribute it to the electrically poweredsensors 52.

The contactless power transmission is carried out preferentially byinductive coupling during rotation of the rotor 50 around the rotationaxis A-A′.

Thus, electrical power is continuously fed to the electrical connection59 and to the electrically powered sensor 52 during rotation of thecentrifuge rotor 50 around axis A-A′, without electrical contact betweenthe rotor 50 and the static parts of the centrifuge 40. The electricalpower is also provided without having to place a battery in thecentrifuge rotor 50.

In this example, the contactless signal transmitter 104 comprises thesame rotatable coil of wires 112 as the contactless power receiver 102and a third electronic card 116 able to transmit electrical signalsconveying measurements made by the electrically powered sensors 52.

The contactless signal receiver 106 comprises the same static coil ofwires 108 as the contactless power transmitter 100, able to receive thesignals which are transmitted without contact by inductive coupling fromthe rotative coil of wire 112 to the static coil of wire 108 and afourth electronic card 118.

Thus, the electrical signals produced by the electrically poweredsensors 52 conveying information on the sensed physical properties ofthe porous sample 10 are transmitted to the contactless signaltransmitter 104 through the electrical connection 59, to the contactlesssignal receiver 106, and then to the control unit 44.

In the present case, the signals include in particular tension andcurrent information measured between each pair of adjacent electrodes80, 82 around the porous sample 10 and advantageously at the sealingsensor 84.

Advantageously, the control unit 44 is able to submit the porous sample10 to a plurality of successive mechanical load levels to establishsuccessive profiles, in particular successive steady state profiles ofsecond fluid content in the porous sample 10.

In particular, it is able to submit the porous sample 10 to a firstmechanical load at a first speed of rotation of the porous sample 44around the rotation axis A-A′ until a first steady state is reached,when the rate of fluid extraction measured by the fluid productionsensor 92 becomes zero.

Then, the control unit 44 is able to submit the porous sample 10 to asecond mechanical load at a second speed of rotation of the poroussample 44 around the rotation axis A-A′ until a second steady state isreached when the rate of fluid extraction measured by the fluidproduction sensor 92 again becomes zero. The second rotation speed isgreater than the first rotation speed.

The control unit 44 is able to recover resistivity and/or conductivitymeasurements measured by the electrically powered sensors 52 at eachsteady state profile j corresponding to successive increasing mechanicalloads, as well as the fluid produced V_(p), until the steady state isreached and to transmit the measured information to the calculator 22.

The calculator 22 is for example a computer having at least a processorand at least a memory containing software modules able to be carried outby the processor.

At each steady state profile j corresponding to a given mechanical load,the calculator 22 is able to assess an initial value of the exponentcoefficient n, and then to calculate, in each region 17, an estimatedsaturation S_(w,est)(i,j) of first fluid, based on the measuredresistivity R_(t)(i,j) measured in the region 17 at the currentmechanical load level j, based on the initial measured resistivity R₀and based on the assessed value of the exponent n.

This calculation is done by using an inverted form of the equationrelating resistivity or/and conductivity to saturation in first fluid,whose exponent coefficient is sought. When Archie's law is used, theinverted equation (1) can be used:

$\begin{matrix}{{S_{w,{est}}\left( {i,j} \right)} = 10^{\frac{\log(\frac{{Rt}({i,j})}{R0})}{- n}}} & (1)\end{matrix}$

The calculator 22 is then able to calculate an estimated volumeV_(1,est) (i,j) of first fluid in the region 17 by multiplying theestimated saturation S_(w,est)(i,j) by an estimated volume of poresV_(po,est)(i) in the region 17.

Advantageously, the estimated volume of pores V_(po,est)(i) is deducedfrom the total volume of pores V_(po) in the whole porous sample 10divided by the number of regions 17 in which an experiment is carriedout.

The total volume of pores V_(po) of the sample is determined for examplevia pycnometry.

Then, the calculator 22 is able to estimate an estimated total volumeVT_(1,est)(j) of first fluid in the porous sample 10 at the steady stateprofile j, by summing all the estimated volumes V_(1,est)(i,j) in thedifferent regions 17.

The calculator is able to calculate a difference D(j) between theestimated volume VT_(1,est)(j) and the measured volume of first fluidVT_(1,mes)(j) inside the porous sample 10 at level j.

The measured volume of first fluid VT_(1,mes)(j) inside the poroussample 10 at level j is equal to the initial volume VT_(1,mes)(j−1) offirst fluid contained in the porous sample 10 before the steady stateprofile j is applied minus the volume V_(p)(j) which has been producedat the level j when the steady state is reached.

The calculator 22 is then able to calculate an objective cost functionwhich is here the sum S of the squares of the differences D(j)².

The objective cost function here has the following form:

$\begin{matrix}{\sum\limits_{j = 1}^{Nj}\left\lbrack {{\sum\limits_{i = 1}^{Ne}\left\{ {\left( 10^{\frac{\log(\frac{{Rt}({i,j})}{R0})}{- n}} \right) \times {V_{{po},{est}}(i)}} \right\}} - {{VT}_{1,{mes}}(j)}} \right\rbrack^{2}} & (2)\end{matrix}$

in which Ne is the total number of regions 17, corresponding to thenumber of adjacent electrode pairs 80, 82 and Nj is the total number ofsteady states profiles corresponding to the mechanical loads to whichthe porous sample 10 is subjected.

Preferentially, Ne ranges from 3 to 15, notably from 7 to 11.Preferentially, Nj ranges from 1 to 10, notably from 4 to 7

Then, the calculator 22 is able to adjust the value of the estimatedrepresentative parameter n and repeat the previous calculation steps tominimize the above-mentioned objective cost function. The representativeparameter n for the porous sample 10 corresponds to the value of n forwhich the objective cost function is minimized.

Similarly, based on the position of each region 17 along the poroussample axis, and on the rotation speed, the calculator 22 is able tocalculate the capillary pressure Pc applied in each region using thefollowing equation:

$\begin{matrix}{{Pc} = {\frac{1}{2} \times \omega^{2} \times {\Delta\rho} \times \left( {r_{s}^{2} - r_{o}^{2}} \right)}} & (3)\end{matrix}$

in which ω is the rotation speed, Δp is the difference of densitybetween the first fluid and the second fluid, r_(s) is the radiusseparating the region 17 from the axis of rotation A-A′, and r₀ is theradius separating the axis of rotation A-A′ from the surface of theporous sample 10 farthest (in drainage) or closest (in imbibition) tothe axis of rotation A-A′.

The calculator 22 is then able to determine a plot of the capillarypressure Pc as a function of the saturation in the first fluid S_(w),calculated in each region from the above mentioned equation (1).

A method for determining a representative parameter of a porous sample10 using the system 12 will now be described.

Initially, a dry porous sample 10 is provided. The volume of poresV_(po) in the sample is evaluated by pycnometry.

The porous sample 10 is then saturated with a first fluid, in particularwith a water-based fluid such as brine.

Then, the porous sample 10 filled with the first fluid is inserted intothe porous sample reception volume 32 of the cell 14.

The cell 14 is introduced in the housing 62 of the centrifuge rotor 50,with the axis X-X′ of the porous sample 10 extending radially withregards to the axis of rotation A-A′ of the rotor 50.

A second fluid is introduced in the upstream chamber 34 located closerto axis A-A′. The second fluid is for example oil, or gas (for exampleair).

The electrically powered sensors 52 are powered by transmitting powerfrom a static part of the centrifuge 40 to the rotor 50 via thecontactless power and signal transmission system 58.

The resistivity R_(o) of the porous sample 10 saturated with the firstfluid is then measured, for example using the tension measured betweenthe end electrodes 80.

Then, the control unit 54 of the centrifuge 40 is activated to actuatethe motor 60 and rotate the rotor 58 jointly with the porous sample 10contained in the cell 14 around the rotation axis A-A′. A firstmechanical load applies on the porous sample 10 due to the centrifugalforce applying on the porous sample 10.

The axis X-X′ of the porous sample 10 extending radially with regard tothe rotation axis A-A′, the second fluid contained in the upstreamchamber 54 progressively penetrates into the porous sample 10 togenerate a profile of saturation in the second fluid which isrepresented schematically with curve 140 in FIG. 5 . In FIG. 5 , therotation axis A-A′ of the porous sample 10 is located on the right ofthe porous sample 10.

The fluid production sensor 92 of the sensing unit 52 is activated tomeasure the rate of fluid extraction from the porous sample 10 collectedin the downstream chamber 36 and the volume of produced fluid V_(p).

In a time period comprised generally between 1 hour and 10 days, asecond fluid content steady state profile establishes in the poroussample 10, when the rate of fluid extraction measured by the fluidproduction sensor 92 becomes zero.

In the steady state profile, the porous sample 10 comprises successiveregions 17 along the longitudinal axis X-X′, the successive regions 17having different local average values of saturation S_(w), in particularincreasing values of saturation in the first fluid S_(w) along thelength of the porous sample 10, taken from the end of the porous sample10 located closer to the axis A-A′ (on the right in FIG. 5 ) to the endof the porous sample 10 located further away from the axis A-A′ (on theleft in FIG. 5 ).

During the measurement, and at the steady stage, power is supplied tothe electrical sensors 52, during rotation of the rotor 50 through thecontactless power transmitter 100, by contactless transmission to thecontactless power receiver 102, and then to the electrical connection59.

The power in particular feeds the electrodes 80, 82, to allowmeasurement of the resistivity in each region 17 located between anelectrode 80 and the adjacent electrode 82, or between two adjacentelectrodes 82.

The measurement of the local resistivity R_(t)(i) carried out betweeneach pair of electrodes 80, 82 or 82, 82 is transmitted from each pairof electrodes 80, 82 or 82, 82 through the electrical connection 59 tothe contactless signal transmitter 104 and without contact to thecontactless signal receiver 106, before reaching the control unit 44.

At each steady state profile j, the calculator 22 assesses an initialvalue of the exponent coefficient n, and then calculates, in each region17, an estimated saturation S_(w,est)(i,j) of first fluid, based on themeasured resistivity R_(t)(i,j) measured in the region 17 at the currentmechanical load level j, based on the initial measured resistivity R₀and based on the assessed value of the exponent n.

This calculation is done by using an inverted form of the equationrelating the resistivity to the saturation, as explained above.

The calculator 22 then calculates an estimated volume V_(1,est)(i,j) offirst fluid in the region 17 by multiplying the estimated saturationS_(w,est)(i,j) by an estimated volume of pores V_(po,est)(i) in theregion 17, as determined above.

Then, the calculator 22 estimates an estimated total volumeVT_(1,est)(j) of first fluid in the porous sample 10 at the steady stateprofile j, by summing all the estimated volumes V_(1,est)(i,j) in thedifferent regions 17 and calculates a difference D(j) between theestimated volume VT_(1,est)(j) and the measured volume of first fluidVT_(1,mes)(j) inside the porous sample 10 at level j, as explainedabove.

The calculator 22 is then calculates an objective cost function which ishere the sum S of the squares of the differences D(j)², as definedabove.

Then, the calculator 22 adjusts the value of the estimatedrepresentative parameter n to minimize the above-mentioned objectivecost function.

The representative parameter n for the porous sample corresponds to thevalue at which the objective cost function is minimized.

The method according to the invention therefore allows a very simple andeffective determination of the exponent coefficient n representative ofa porous sample 10 by applying successive mechanical loads in acentrifuge 40 and by measuring the produced first fluid volume and theresistivities or/and conductivities in different regions 17 along theporous sample 10.

The latter measurement is carried out continuously when operating themethod, advantageously by powering the electrically powered sensors 52in a contactless manner and by receiving data from the electricallypowered sensors 52 in a contactless manner.

The method does not require a handling of the porous sample 10 duringthe experiment, even if successive increasing levels of mechanical loadare applied to the porous sample 10. The porous sample 10 remains in thecentrifuge 40 during the whole experiment.

There is no need to use external techniques to determine the saturationin first fluid in the porous sample 10.

The timeline and cost for carrying out the method according to theinvention are therefore greatly reduced, while still obtaining a verywide range of data.

In a variant, only one level of mechanical load is applied to thesample.

In another variant, the information generated by the electricallypowered sensors 52 carried by the rotor 50 is transmitted by a wirelesstransmitter independent of the contactless power transmission to thesensors 52.

In a further variant, the method is carried out using a centrifuge 40having a rotatable electrical connector, having for example brushes,between the rotor 50 and the static part of the centrifuge 40.

In another variant, the method is carried out without necessarilyestablishing a steady state profile of second fluid content in theporous sample 10. The established profile is for example a transientstate profile obtained by applying the mechanical load, before a steadystate profile is obtained.

1. A method to determine a representative parameter of a porous samplein an equation relating a resistivity or/and conductivity of the poroussample, with a saturation of the porous sample in a first fluid, themethod comprising: providing a porous sample containing a first fluid;feeding a second fluid in the porous sample and establishing at leastone first profile of second fluid content in the porous sample byapplying a first mechanical load; measuring a resistivity or/andconductivity in a plurality of regions having different second fluidcontents in the porous sample; measuring a volume of first fluidproduced from the porous sample when establishing the at least one firstprofile; calculating a measured total volume of first fluid remaining inthe porous sample after establishing the at least one first profile; andrepeating: (i) determining an estimated local volume of first fluidcontained in each region from the resistivity or/and conductivitymeasured in the region and from an estimated value of the representativeparameter; (ii) calculating an estimated total volume of first fluid inthe porous sample from each estimated local volume of first fluidcontained in each region; (iii) modifying the estimated value of theestimated representative parameter to minimize the difference betweenthe estimated total volume and the measured total volume, therepresentative parameter of the porous sample being the estimated valueof the representative parameter minimizing said difference.
 2. Themethod according to claim 1, wherein the local estimated volume of firstfluid contained in each region is calculated from a saturation in firstfluid obtained from an inversion of the equation relating theresistivity or/and conductivity and the saturation in a first fluid inthe porous sample, using the resistivity or/and conductivity measured inthe region and using the estimated value of the representativeparameter.
 3. The method according to claim 1, wherein determining thelocal estimated volume of first fluid contained in each region comprisescalculating a local volume of pores in each region.
 4. The methodaccording to claim 1, comprising after measuring the resistivity or/andconductivity in a plurality of regions having different second fluidcontents in the porous sample: establishing at least one subsequentprofile of second fluid content in the porous sample by applying asubsequent mechanical load different from the first mechanical load;measuring in the plurality of regions a subsequent resistivity and/orconductivity and a subsequent volume of first fluid collected in theestablishment of the subsequent profile, calculating a subsequentremaining volume of first fluid in the porous sample, the repeatingcomprising, after each subsequent profile has been establishedestablished: determining an estimated volume of first fluid contained ineach region from the resistivity or/and conductivity measured in theregion and from the estimated value of the representative parameter;calculating an estimated total volume of first fluid in the poroussample from each determined estimated local volume of first fluid volumecontained in each region after each the subsequent profile has beenestablished; modifying the value of the estimated representativeparameter to minimize the difference between each estimated total volumeand the corresponding measured total volume after the subsequent profilehas been established, the representative parameter of the porous samplebeing the estimated representative parameter minimizing said differenceafter each subsequent profile has been established.
 5. The methodaccording to claim 4, wherein the first profile and/or the at least onesubsequent profile are established in a centrifuge.
 6. The methodaccording to claim 5, wherein establishing the first profile and the atleast one subsequent profile is carried out without removing the poroussample from the centrifuge.
 7. The method according to claim 1, whereinminimizing the difference between the estimated cumulated volume and themeasured volume comprises calculating a cost function according to thefollowing equation:$\sum\limits_{j = 1}^{Nj}\left\lbrack {{\sum\limits_{i = 1}^{Ne}\left\{ {\left( 10^{\frac{\log(\frac{{Rt}({i,j})}{R0})}{- n}} \right) \times {V_{{po},{est}}(i)}} \right\}} - {{VT}_{1,{mes}}(j)}} \right\rbrack^{2}$in which Nj is a total number of steps of establishing a profile ofsecond fluid in the porous sample, Ne is the total number of regions inthe porous sample, R_(t)(i,j) is a resistivity in a i^(th) region aftera j^(th) step of establishing a profile of second fluid in the poroussample, R0 is an initial resistivity in a i^(th) region before feedingthe second fluid in the porous sample, n is the estimated representativeparameter, V_(po,est)(i) is an estimated volume of pores in the i^(th)region, VT_(1,mes)(j) is a measured volume of first fluid inside theporous sample 10 after a j^(th) step of establishing a profile of secondfluid in the porous sample.
 8. The method according to claim 1, whereinmeasuring the resistivity or/and conductivity in a plurality of regionshaving different second fluid contents in the porous sample is carriedout while the first mechanical load is applied.
 9. The method accordingto claim 1, wherein the equation relating the resistivity or/andconductivity and the saturation in a first fluid is chosen amongArchie's law, Waxman-Smits's law, Poupon-Leveaux's law, Simandoux's law,Clavier-Coates Dumanoir's Dual-Water law and/or the effective Spalburg'smedium model law.
 10. The method according to claim 1, wherein the firstfluid is a water based fluid, the second fluid being an oil based fluid.11. The method according to claim 1, wherein the porous sample is aformation sample.
 12. The method according to claim 1, comprising,before feeding a second fluid in the porous sample and establishing afirst profile of the second fluid, measuring a resistivity or/andconductivity in the plurality of regions when the porous sample onlycontains the first fluid.
 13. A system for determining to determine arepresentative parameter of a porous sample in an equation relating theresistivity or/and conductivity of the porous sample, with a saturationof the porous sample in a first fluid, the system comprising: a cellreceiving a porous sample containing a first fluid; a feeder configuredto feed a second fluid in the porous sample and a first mechanical loadapplier configured to establish at least a first profile of second fluidcontent in the porous sample; resistivity or/and conductivity measuringsensors for measuring configured to measure a resistivity or/andconductivity in a plurality of regions having different second fluidcontents in the porous sample; a fluid production sensor configured tomeasure a volume of first fluid produced from the porous sample whenestablishing the first profile; a calculator configured to determine ameasured total volume of first fluid remaining in the porous sampleafter establishing the first profile; the calculator being configured torepeat: (i) determining an estimated volume of first fluid contained ineach region from the resistivity or/and conductivity measured in theregion and from an estimated value of the representative parameter; (ii)calculating a estimated total volume of first fluid in the porous samplefrom each determined estimated local volume of first fluid volumecontained in each region; (iii) modifying the value of the estimatedrepresentative parameter to minimize the difference between theestimated total volume and the measured total volume, the calculatorbeing configured to determine the representative parameter of the poroussample as the estimated representative parameter minimizing saiddifference.
 14. The system according to claim 13, wherein the firstmechanical load applier configured to establish a first profile ofsecond fluid content in the porous sample comprises a centrifuge havinga static part and a rotor rotatable with regards to the static part, theresistivity or/and conductivity measuring sensors being electricallypowered sensors carried by the rotor, the centrifuge comprising acontactless power and signal transmitter between the rotor and thestatic part of the centrifuge.
 15. The method according to claim 3,wherein determining the local estimated volume of first fluid containedin each region comprises calculating a local volume of pores in eachregion from a global volume of pores determined in the porous sample.16. The method according to claim 1, wherein the first fluid is brine.17. The method according to claim 11, wherein the porous sample is arock sample.